Critical scaling of a two-orbital topological model with extended neighboring couplings

Extended-range models are the interesting systems, which has been widely used to understand the non-local properties of the fermions at quantum scale. We aim to study the interplay between criticality and extended range couplings under various symmetry constraints. Here, we consider a two orbital Bernevig–Hughes–Zhang model in one dimension with longer (finite neighbor) and long-range (infinite neighbor) couplings. We study the behavior of model using scaling laws and universality class for models with Hermitian, parity-time (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{P}\mathscr{T}$$\end{document}PT) symmetric and broken time-reversal symmetries. We observe the interesting results on multi-criticalities, where the universality class of critical exponent is different than the normal criticalities. Also, the results can be generalized by considering the interplay between criticalities and different symmetry classes of Hamiltonian. Also, with the introduction of extended-range of coupling, there occurs different criticalities, and we provide the analogy to characterize their universality classes. We also show the violation of Lorentz invariance at multi-criticalities and evaluation of short-range limit in long-range models as the highlights of this work.

As a generalized platform for our analysis, we adopt a one dimensional version of a two orbital Bernevig-Hughes-Zhang (BHZ) model 24 .Originally, BHZ model is an efficient proposal for the realization of quantum spin Hall effect in two dimension 25 .Here, Z 2 topological insulator is realized in a quantum where HgTe is sandwiched between crystals of CdTe.There are numerous works which efficiently explains the topological properties of model both experimentally and theoretically 2,5 .Here, we consider the one dimensional reduced BHZ model to explain the interplay of symmetry and criticality with the extended-range of coupling, and also to characterize the topological criticalities especially when two criticalities meet each other, i.e., multi-criticalities.

Model Hamiltonian
We consider BHZ model in 1D with extended-range of couplings.The model consists of spinless non-interacting fermions in the s and p x orbitals, where the interactions like p x ± ip y and the intra-orbital hopping i.e. hopping between s ←→ p x of same unit cells are excluded.The generalized 1D BHZ Hamiltonian is given by 24 The hopping strength between s j ←→ p j+1 (s j+1 ←→ p j ) and s j ←→ s j+1 (p j ←→ p j+1 ) is t ps (−t ps ) and t p (t s ) respectively.Also, the terms s and p x orbital consists of on site potentials ǫ s and, ǫ p respectively.Here the term j symbolizes the lattice site, which can take a larger value L. If the coupling occurs only among the nearest neighbors, it is referred as short-range, whereas, if it occurs among ith and i + l th sites, it is called extended-range coupling 17,26 (Fig. 1).
After Fourier transformation, the model can be written in the Bloch Hamiltonian as, with with j = 0, 1, 2, 3 .Here σ 0 , σ x,y,z and χ 0,x,y,z are identity matrix, Pauli spin matrices and winding vectors respec- tively.Before studying the system in detail, we need to understand the Hermitian behavior of the model.For this purpose, we substitute ǫ p = −ǫ s = ǫ and t s = t * s = t p = t * p = t into Eq.(1).Under standard non-interacting conditions, the Hamiltonian becomes with It is to be noted that for r → ∞ the series involving cos(kl) l α and sin(kl) l α terms give rise to polylogarithmic functions 11,26 .Here, at first we consider finite number of interacting neighbors (r) and analyze the scaling − t s l α s † j s j+l + t p l α p † j p j+l + t ps l α s † j p j+l + H.c.
(2) properties through momentum space characterization.The model is called as isotropic, as the coupling parameters decay with the same strength (i.e., t = t ps ).Basically, the symmetric properties of the models are unaltered with the addition of extended interacting neighbors.The tenfold classification includes discrete symmetries such as time reversal symmetry (TRS), particle-hole symmetry (PHS) and chiral symmetries (CS).The TRS is an anti-unitary operator, represented as the product of unitary ( U ) and complex conjugation ( K ), i.e., T = U K .For the spinless systems , T 2 = 1 and symmetry is said to be obeyed if the Hamiltonian commutes with the operator, i.e., T , H = 0 .PHS sym- metry is another non-unitary operator with C 2 = 1 which anti commutes with the Hamiltonian as {C , H} = 0 .This signals the transformation between electron and holes under the given range of energy.The product of TRS and PHS gives the CS, which is equivalent to the sub-lattice symmetry in the Hermitian systems.This is an anti-unitary operator with {S , H} = −H , which reveals the properties of a symmetric spectrum.The general energy spectrum is given by The topological invariant equation is given by which yields integer for topological state and W = 0 for trivial states respectively 27 .Here we calculate different critical exponents using momentum space characterization to understand different criticalities.The universality class can be constructed using different exponents such as dynamic (z), localization ( ν ), crossover (y), suscepti- bility ( γ ) and canonical ( α * ) critical exponents as bellow.
Dynamical and crossover critical exponent: This critical exponent can be calculated by expanding Eq. ( 5) around the gap closing points k 0 as, where (g − g c ) 2y signals the distance in the parameter space, y is the crossover and z is the dynamical critical exponents respectively.Here y and z determine the gap opening/closing and nature of dispersion, which can be calculated as

Localization critical exponent:
The topological phase possesses the localization of zero energy edge modes, which are protected by the topology of bulk Bloch electronic states.In the open boundary condition, the Bloch Hamiltonian (Eq.4) can be written as with Here ûn = (u n (1), u n (2), . . .u n (N)) and follows the Hermiticity and time reversal symmetry through the relation Here n is the lattice index and the Fourier transformation gives the form k ψ † k H(k)ψ k which can also be written in the form of Eq. ( 4).The ratio of amplitudes of the nth localized Eigen state ( ψ n ) to the first ( ψ 0 ) is given by 28 , These zero energy states are localized at the edge of the chain which are ensured by the condition e ik 0 = − δg A 2,4 .For zero energy state, k 0 is the complex number which leads to ψ n = e ik 0 (n−1) , where n is the system size.Through Eq. ( 7), we can calculate k 0 as, where the dominating term among A 4 and A 2 decides the value of k 0 , which finally leads to (5) , where ν is the localization critical exponent.Susceptibility critical exponent γ : This critical exponent can be extracted by the integrand of Eq. ( 6) which shows divergence as one approaches criticality 27,[29][30][31] .This is also called curvature function (CF), and can be written in the Ornstein-Zernike form (which is traditionally famous for relating different correlation factors with each other).
Here we characterize the critical point as high symmetry (HSP) and non-high symmetry points (non-HSP) based on the behavior of CF around the gap-closing point k 0 .If the CF exhibits a symmetric nature of gap- closing in the Brillouin zone ( k 0 = −k 0 ), then it is called HSPs.In such case CF acts as an even function, i.e., F(k 0 + δk, g) = F(k 0 − δk, g) around k 0 (Here g is the set of parameters).As we approach the critical point from one side ( g + → g c ), the CF shows a Lorentzian peak around k 0 and flips as we pass criticality ( g c → g − ).The Lorentzian peak becomes non-analytic at k = k 0 which results in the ill-defined winding number at criticality.On the other hand, non-HSP are the points which do not exhibit even nature around k 0 but shows non-analytic Lorentzian peak at k 0 17 .To obtain the critical exponents, we expand the CF around gap closing points, where the coefficients of δk 2 and δk 4 decide the nature of correlation ( ξ ).For our 1D system, we consider Berry connection as our CF, whose behavior is characterized by the critical exponents ν and γ .i.e. 27,32 , Here γ and ν are the susceptibility and localization critical exponent, which signal the exponent associated with the Berry connection 33 .
Ground state energy: For a system with real energy spectrum, each energy interval dE contains |k| d occupied states, where d is the spacial dimension of the system.The ground state energy (GSE) indicates TQPTs with the singularities in the parameter space ( ω singular ), where the order of the its derivative is related to the order of transition.Thus a relation can be established between GSE and other critical exponents as, where (d + z) is the effective dimension 34 .This concept leads to the famous relation known as Josephson's hyper-scaling relation which relates the effective dimensionality with the order of phase transition 7 .(Here α * is called canonical critical exponent and not to be confused with decay parameter α ).To analyze the order of transition, we perform the scaling of the singular part of the GSE 34 .As the GSE is sensitive to system size, it needs a multiplication of every length of GSE by localization factor 34 i.e., which gives the effective dimension of the system.
Thus, the set of critical exponents yield the universality class which is an efficient to categorize different criticalities.In the following section, we considers different examples to understand the interplay between criticality and extended range coupling under various symmetry constraints.

Results
Here we present three different ranges of coupling, i.e., short-range, extended-range and long-range for Hermitian, broken TRS and P T symmetric respectively.

Hermitian condition
Under the standard Hermitian conditions, we obtain the Hamiltonian, Here the Hamiltonian follows TRS, PHS and CS as shown in Table 1, which results in AIII symmetry class of AZ classification.In general, when the Hamiltonian obeys TRS, PHS and CS symmetries, a topological superconductor falls under BDI class and makes a 1D model as a Z topological invariant .But in a topological insulator (like SSH), the chiral symmetry is enforced and other symmetries are accidental.This is just a matter of choice and does not alter the physics of criticality.(This factor becomes significant, when one prefers to characterize the topology of the system purely based on the symmetric behavior.Here we just restrict ourselves for the critical characterization).( 13) The operators T is nothing other than the complex conjugate K .The operator C is the combination of two terms, i.e., C = U c K .Here U c is a unitary matrix and for the current case it yields σ x .The operator T C is the combination of above operators.Here, we consider three different ranges of coupling to understand the behavior of criticality, i.e., short-range, extended-range and long-range.

Short-range
As a first case, we consider the coupling up to first interacting neighbor only, i.e. short-range ( α → ∞ ).The winding vectors are given by Here we consider t ps = 1 for the simplicity.The phase diagram is given by Fig. 2a, where the criticality occurs with linear dispersion at ǫ = 2t and ǫ = −2t for k = 0 and π respectively (Fig. 2b).Here we obtain topological phase for ǫ < 2t and trivial phase for ǫ > 2t.
By using Eqs.( 20) and ( 13), we get CF, which is a HSP around k = 0 and π .The GSE shows the singular- ity for the second order derivative signaling a second order phase transition where the critical exponents are presented in Fig. 2b-e.

Extended-range
We consider a simple extended-range model having second nearest neighbor ( r = 2 ) coupling with t = t ps condition.The phase diagram is given by Fig. 3a.The quasi-energy dispersion at k = 0 is given by Eq. ( 54), which remains linear for all the values of α as the term A 2 = 2t 1 − 4 2 α always dominates over A 4 .Thus, at k = 0 , the dynamical critical exponent is z = 1 irrespective of the value of α .Due to the effect of multi-criticality, the nature of dispersion is quadratic ( z = 2 ) around k = π at α = 1 as the term . This is a nice example of the breaking of Lorentz invariance (for α = 1 ), the instances of which are the area of curiosity in the condensed matter systems (For analytical calculations, please refer "Method" section).We explain on this more in the following section.
Similarly, the dominating term among A2 and A4 determines the exponent ν .At k = 0 , the term A2 > A4 , and contributes majorly to the divergence of localization ξ by yielding ν = 1 for all the values of α .In the vicinity of multi-critical point, the behavior is quite different.At k = π , the term A2 > A4 for all values except α = 1 , which gives ν = 1 in this regime.At α = 1 the term A4 > A2 and yields ν = 1/2 .Thus, the scaling law zν = 2 × 1/2 = 1 also remains valid at multi-criticality (Fig. 3b-e).( 20) Table 1.Symmetry operation of a Hermitian model.Here TRS, PHS and CS represent time-reversal, particlehole and chiral symmetries respectively.To understand the order of phase transition, we consider Josephson hyper-scaling relation Eq. ( 17) as signaling a fractional order of transition at multi-critical points.To understand the nature of scaling, we substitute into Eq.( 18), giving Thus, the GSE scales similarly for normal criticality and at multi-critical point even though both belong to different universality classes.On the other hand, both at criticality and multi-criticality, the GSE shows non-analyticity for the second order derivatives (not shown here).This shows that, the bulk topological transitions are the second order transitions even at multi-criticality.At multi-criticality ( k = π, α = 1, ǫ = 1 ), the two critical lines intersect each other, which separate at least three topological phases.This kind of intersection results in fixed point configuration where the CF fails to exhibit the even nature around the HSP.Thus, at the multi-criticality, CF does not exhibit non-analyticity, instead shows a curve with constant height and varying width 17 .This kind of behavior is absent in other transitions.Interestingly, the scaling of CF gives γ = 1 for all criticalities including multi-criticality (Fig. 3d).

TRS
The further increase in the neighbor coupling produces different phase diagram and corresponding critical lines.As one increases the coupling neighbors (for all r > 2 ), there occurs a staircase of topological transitions only among even-to-even and odd-to-odd winding numbers 17 .Interestingly, we observe multi-critical behavior, only in the presence of even neighbor couplings.Thus, all the topological transitions (irrespective of neighbors) falls to single universality class except multi-criticality (Table 2).

Long-range
When there are infinite number of neighboring coupling, the pseudo-spin parameters takes polylogarithmic behavior as 11,17 , The energy (Eq.5) gap closing occurs for k = 0(α > 1) and π(∀α) .There occurs a diverging energy spectrum around k = 0 for α ≤ 1 , due to the polylogarithmic nature of χ y 11 .Thus, one can get the phase diagram as (Fig. 4a)  17,26 .Due to the non-analytical nature of the CF, the topological invariant is ill-defined for α < 1 .However, here the CF shows an removable singularity, where the singularity can be integrated out and the WN yields fractional values (Fig. 4a).Expansion of polylogarithmic functions is given by 35 , For the detailed study, we can expand the polylogarithmic function around k = 0 as Thus the pseudo-spin vectors are monitored by the gamma function which depends on α .Here, α can influence the properties of the phase diagram, gap closing, energy dispersion and Fermi velocity.The energy dispersion is Table 2.A comparison of universality class of critical exponents between the short-range, extended-range and long-range models.Here IL, D and H represent ill-defined, divergent and higher order quantities.given by the Eq. ( 5) with pseudo-spin vectors explained in Eq. (23).To understand the polylogarithmic nature, we express in terms of Eq. ( 25) and obtain the dispersion as 11 , where A, B and C are constants.Thus, as k → 0 the energy vanishes only for α > 1 while as k → π , the energy vanishes for all values of α .Event though the energy vanishes the derivatives fail to vanish in the limit k → 0 for 1 < α < 2 which reflects in corresponding critical exponent.By substituting Eq. ( 25) into Eq.( 5) and expanding the energy dispersion equation around the gap closing point k 0 , up to a leading order, we get Eq.( 7) with coefficients for k = 0 and π respectively.For k = π , the A 2 term dominates for all values of α , which ensures a linear disper- sion i.e., E(k 0 , g c ) ∝ k .For k = 0 , the A 2 term shows a divergence with a factor Ŵ[1 − α] .Thus, α > 2 region shows a linear dispersion with E(k 0 , g c ) ∝ k , while the 1 < α < 2 region shows a dispersion with z < 1 and y = 1 (Fig. 4b,d).Our results are in agreement with the Ref. 14 , which deals with a similar model through field theoretical methods.
The CF is given by Eq. ( 13).To understand the α dependent non-analyticity, we substitute Eq. ( 25) into Eq.( 13) which is expanded around k 0 the first leading order and after few steps of simplification, we get the CF in Ornstein-Zernike form as There are three possible cases, Here we find CF in Ornstein-Zernike form only for α > 2 around k = 0 and for all α around k = π , which yield γ = ν = 1 in these regions (Fig. 4c,e).Thus, The long-range model attains the universality class of short-range for the range α > 2 , which can be considered as the short-range limit of the model.

Hermitian case with broken TRS
The Hermitian model preserves the time-reversal symmetry and it can be broken with the introduction of a phase difference in the hopping term as t s → |t|e iφ , where t is a real quantity.This results in a complex hopping and the model shifts from AIII to D symmetry class in AZ symmetry classification 26 , where PHS is preserved (Table 3).
The breaking of TRS results in a gapless region along with topological and trivial phases.The analysis shows that φ takes the values in the regime [0, π/2] which produces different phase diagrams 26 .The Hamiltonian is given by, After Fourier transform the Hamiltonian can be written as with The energy dispersion is given by ( 26) t ps l α sin(kl).
Table 3. Symmetry operation of a Hermitian model with broken time-reversal symmetry.Here TRS, PHS and CS represent time-reversal, particle-hole and chiral symmetries respectively.

TRS
Vol.:(0123456789) www.nature.com/scientificreports/ The curvature function is not directly involved in defining the topological invariant or number of edge modes.But the topological index can be verified with the presence or absence of the edge modes.The phase diagram consists of gapped and gapless phases which can be determined by the relation 26 The quantity sign(η) = ±1 signals the gapped and gapless phases respectively.The transition from gapped to gapless phase represents the topological transition and the corresponding critical properties can be measured.
The topological phase occurs in the regime 26 When the angle φ is independent of the l, i.e., φ l = φ the relation becomes We observe two kind of gapless regions here.The gapless region because of the momentum vector ( k = 0, π ) and the phase factor φ .for a finite value of φ , the energy spectrum yields an elliptic equation χ 2 0 = χ 2 z + χ 2 y .The model obeys TRS only for 0, π .Hence for all other finite values of φ , the χ 0 term produce a finite value and a 2D elliptic region is produced with gapless spectrum and an ill-defined topological index.This region contains a horizontal boundary (in t ps − ǫ parameter space) with t ps = ± sin(φ) condition and a vertical boundary with ǫ = ± cos(φ) condition.As the number of interacting neighbors increase, vertical boundary gets the multipli- cative terms.A hemispherical cap is added to the vertical boundaries, which touch the criticality k = 0, π .The region outside this elliptical structure contains integer topological invariant and the criticality k = 0, π separates topological and trivial phases.
The interface of these two gapless regions constitutes a multi-critical point, which is unique than other criticalities.Thus, we consider three ranges of couplings to understand the behavior of these criticalities.Here, our interest is to explore the behavior of multi-criticality and for our purpose we consider a single parameter space φ = 3π/10 .The further variation of φ alters the phase diagram 26,36 , but the general behavior of criticalities fol- low the similar pattern.

Short-range
With the coupling up to first nearest neighbor, we write Eq. (30) as The energy dispersion is given by Eq. (31) and criticality occurs at The gap closing occurs at k = 0, π with criticality ǫ = ±2t cos(φ) .The parameter φ takes the value between 0 and π/2 where the horizontal and vertical boundaries are determined by ǫ = ±2t cos(φ) and t ps = ±t sin(φ) respectively.The ends of the gapless region touches the points (t ps , ǫ) = (0, ±2) , which creates an elliptical gap- less region as shown in Fig. 9a.
The energy dispersion shows a linear spectrum at ǫ = ± cos(φ) while at multi-criticalities show linear dis- persion only at one side.(Fig. 5b).The broken TRS leads to the breaking of even nature of spectrum around the gap closing pint.Thus, we get z = 1 and z = 3 around left and right sides of gap closing point respectively (Fig. 5d).The crossover critical exponent yields linear curve for both left and right side of the multi-critical point (Fig. 5f).The localization of the edge mode can be determined by Eq. ( 36), which yield linear spectra around multi-criticality (Fig. 5e).

Long-range
Due to the long-range effects, Eq. ( 30) becomes Here we consider the value of φ independent of index 'l' , hence they are not expressed in terms of polylogarithmic functions.The criticalities occur at ǫ = −2t cos(φ)Li α (±1) producing the phase diagram Fig. 7a.The energy dispersion can be obtained by substituting Eq. ( 39) into Eq.( 31) as shown in Fig. 7b,c.At k = 0 , the dispersion is divergent with ill-defined critical exponent.For the range 1 < α < 2 and α > 2 , we observe a square root ( z = 1/2 ) and linear ( z = 1 ) dispersion respectively (Fig. 7b1).For k = π , dispersion remains linear for values of α with z = 1 (Fig. 7b2).The crossover critical exponent remains same (y = 1) for all criticalities (Fig. 7c1-c2).The CF can not be expressed in terms of Ornstein-Zernike form, hence the critical exponents γ can not be expressed with current methodology.But through the relation Eq. ( 54), we obtain ν = 1 for α > 2 regime (Fig. 7d1-d2).A comparison of all critical exponents can be given by Table 4 PT symmetric non-Hermitian models In general, the imbalance in the hopping or the addition of complex potential leads to a non-Hermitian behavior of the system, with a complex energy spectrum.With the addition of P T symmetry, the space-time reflection can be preserved, where the models can be treated equivalent to Hermitian counterpart irrespective of the non-Hermitian signatures.In our case, non-Hermiticity can be introduced into the model with a non-reciprocity parameter δ in the hopping term as (39) Thus the forward hopping ( s j → p j+l ) and backward ( s j ← p j+l ) becomes t − δ and t + δ respectively.Thus, in the region t ps > |δ| the Hamiltonian produces the real spectra and t ps < |δ| produces complex energy spectrum respectively.The real part contributes to the filling of Fermi level, while the imaginary part contributes to the phase.Hence, we obtain the information of dynamical and crossover critical exponents from the energy dispersion.Through the analysis of bi-orthonormal vectors, the complex Berry phase can be constructed, and the integrand can provide the information of the curvature function.The curvature function contains the complex form, where the real part contributes to the argument and the imaginary contributes towards amplitude.The non-reciprocity in hopping produces a coalescence of Eigen vectors, which results in non-Hermitian skin effect and breaking of bulk-edge correspondence.However, here we only concentrate on the critical properties under different coupling ranges.
Symmetry properties: The non-reciprocity term changes the Hamiltonian from Hermitian to non-Hermitian without altering the TRS and PHS.In, non-Hermitian systems, the CS and sub-lattice symmetries are different unlike Hermitian systems 24 .In the current model, the SLS is broken while CS is obeyed (Table 5).In addition to this, the combination of parity and time reversal P T symmetry obeyed for the region t ps > |δ| , which gives rise to a real energy spectrum.
Complex Berry phase: Under Hermitian case, the Berry phase is a real quantity and remains quantized under certain symmetries by reflecting the topological order of the system.In case of non-Hermitian systems, the periodic table of symmetry is different and the structure of the topological invariant may not behave similar to that of Hermitian systems.This creates a necessity to generalize the structure of topological invariants to a wide spectrum of non-Hermitian systems, which leads to the concept of complex Berry phase [37][38][39] .The geometry is constructed in the bi-orthonormal basis, where the structure behaves similar to that of Hermitian Berry phase.The complex Berry phase can be obtained through the Fourier transform of Eq. ( 40) as with (40) Table 4.A comparison of universality class of critical exponents between short, extended and long-range of Hermitian model with broken time-reversal symmetry.Here IL and D represent ill-defined and divergent quantities.
Table 5. Symmetry operation of a P T symmetric non-Hermitian model.Here TRS, PHS, CS and SLS represent time-reversal, particle-hole, chiral and sub-lattice symmetries respectively.

TRS
Vol.:(0123456789)The dispersion remains real for the P T symmetric regime, where we consider the spectra to calculate the criti- cal exponents.For the P T broken regime, the spectra become complex, and we consider the absolute spectra to calculate the critical exponents.Sometimes, the real spectra produces different critical exponents around the single gap closing point, based on the symmetry behavior.To avoid such ambiguity, we consider the absolute spectra which may produce different critical exponents at some parameter space 34 .To understand the geometric phase and associated critical exponents, we introduce bi-orthonormal basis vectors as 24 Here κ is independent of k with with β * 1 β 2 = 1 2 (1 − cos(θ)) which are periodic in k.At exceptional point we get � ± (k)|ψ pm (k)� = 0 .The complex Berry phase is given by 37,40,41 Complex Berry phase gives the Hermitian equivalence of the geometric phase by deploying the left and right eigenvectors.The integrand of Eq. ( 46) can be used as a curvature function to determine critical exponents.Here, we consider three different ranges of couplings to understand the behavior of the system.Due to the structure of energy dispersion (Eq.43), for each range of couplings, the criticality condition remains similar to that of their Hermitian counterparts.Hence the phase diagram remains similar in Hermitian and P T symmetric models, but the symmetry behavior changes for different parameter spaces.

Short-range
Here, the winding vectors in rotated basis is given by, The energy dispersion and complex Berry phase is given by where the spectrum preserves P T symmetry for t > δ .At t = δ there occurs a transition from P T symmetric to P T broken phase and the spectrum becomes complex for δ > t .The universality class of critical exponents remain similar to that of Hermitian system under P T symmetry, and we consider the P T broken phase and P T transition point for critical analysis.
At t ps = δ , the dispersion becomes linear and the crossover exponent shows a linear fitting.The exponents γ and ν can be calculated through Eqs.(54) and (54) respectively, which yield linear fittings (Fig. 8a1-d1).For the value t ps < δ the Hamiltonian becomes P T broken producing the non-Hermitian skin effect.The energy spectrum yields the complex values, where the exponents may differ for real and absolute spectra.Here we consider only absolute spectrum and get the universality class z = y = ν = γ = 1 respectively (Fig. 8a2-d2).

Long-range
Here, the winding vectors in rotated basis is given by, The energy dispersion and complex Berry phase is given by ( 48)  For t ps < δ , the universality class of critical exponents yield z = y = ν = γ = 1 in the regime α > 2 .Here we consider absolute spectrum, which results in the domination of non-Hermitian properties (Fig. 10a2-d2).At the point of P T symmetry breaking ( t ps = δ ), the localization exponent vanishes as the localization length becomes slop less (Fig. 10a1-d1).In the regime 1 < α < 2 , the dynamical exponent yields fractional values with crossover exponent y = 1 .The susceptibility and localization exponents are ill-defined in this regime (not shown here).A comparison of critical exponents are presented in Table 6 Discussion

Manifestation of Lorentz invariance at multi-critical points
As per the scaling law, the linear dispersion of energy (at gap closing point/Dirac points) always preserves the Lorentz invariance .Dynamical critical exponent (z) is the quantity which reflects the zero mass condition, including the information about the Lorentz invariance 42 .At Fermi surface, the energy dispersion is similar to relativistic form (which is a Lorentz invariant), i.e., E 2 = c 2 p 2 + m 2 c 4 , which can reduce to massless case (under criticality).Hence, it becomes a linear equation which connects the energy and momentum ( E 2 = c 2 p 2 ).
• In longer-range model, due to the multi-criticality effect, the dispersion relation acquires some correction term i.e., E 2 = c 2 p 2 + m 2 c 4 + Ap 4 , where A is a constant 43 .This results in the violation of Lorentz invariance and gives rise to quadratic dispersion of energy z = 2 .The velocity of quasi-particles around k = π criticality is given by, where A, B and C are constants.For criticality, the velocity vanishes as g → g .But for multi-criticality, the velocity becomes indeterminate as α → 1. • In long-range model, the dimensionality of the system may change depending on the strength of coupling 15,16 .
In the 1 < α < 2 region, the Lorentz invariance breaks and the quasi-particles near the criticality feel the effective dimension.The velocity of quasi-particles around k = 0 criticality is given by, where A, B and C are constants.For the limit α > 2 , the velocity vanishes as k → 0 .When z = 1 , the exited quasi-particles near the gapless states feel a fixed speed.For z > 1 , the speed is not fixed and for z < 1 , the exited quasi-particles possess very small momenta and contains no upper limit to the speed 44,45 .In long-range models, the conservation of Lorentz invariance in the low excitation spectrum validates the correlation exponents through Ornstein-Zernike equations.As the Lorentz invariance is broken in the region 1 < α < 2 , the CF fails to express the correlation and susceptibility exponents.
(49) Through Table 7, we observe that, the quadratic dispersion occurs only for the systems with even number of couplings.This observation holds true for higher number of couplings also.

Conclusion
Multi-criticalities in the longer-range (finite neighbors) models are the combinations of criticalities of different nature.The localization property and the behavior of topological invariant at criticality ignites the curiosity in this area.This kind of multi-criticality also witness different possible transitions which can also be recognized by the universality class of critical exponents.It also helps to categorize the criticalities to recognize the breaking of Lorentz invariance.The is consistent with multi-criticalities with different symmetry behaviors.At some stage, multi-criticalities in an extended model with broken time-reversal symmetry exhibits an exponent with z = 4 , which is similar to the flat band structures.The methodology is extended to long-range models (infinite neighbors), where the universality class of critical exponents recognizes 1 < α < 2 as Lorentz violated region and α = 2 as short-range limit.To summarize, we present a theoretical study of criticality for a short, extended and long-range topological chain under different symmetry conditions.Instances of interplay of criticality and extended-range under symmetry constrains are rare in literature, and we hope our work will be interesting in understanding such systems.Table 7. Table of critical exponents for longer-range two orbital model with t = 0.5 obtained by an analysis of Eqs. ( 14) and (7).Throughout the analysis, with the even number of neighbors, 1st TQCL k = 0 exhibits same universality class i.e., (z = 1, ν = 1) .But 2nd TQCL k = π shows a breakdown at α = 1 i.e., (z = 1, ν = 1) to (z = 2, ν = 1/2) .For the odd interacting neighbors, the universality class remains same for both k = 0 and k = π.

Figure 1 .
Figure 1.Schematic representation non-interacting extended-range BHZ model in one dimension.Colored rectangles represent the unit cell, with red (yellow) circles representing s ( p x ) orbitals.The neighboring coupling decays with power law for the lth neighbor.

Figure 3 .
Figure 3. (a) Phase diagram of extended-range with two interacting neighbors.(b) Dispersion giving z = 2 at multi-criticality.The inset represents linear dispersion at normal criticalities.(c-e) crossover, susceptibility and localization critical exponents around k = π for multi-criticality (blue) and normal criticality (red) respectively.

Figure 4 .
Figure 4. (a) Phase diagram of long-range model.(b-e) Dynamical, susceptibility, crossover and localization critical exponents for Hermitian long-range model.

Figure 5 .
Figure 5. (a) Phase diagram of Hermitian model with broken TRS with φ = 3π/10 .The colored gapless region is ill-defined and topological quantities can not be defined in this region.(b) Energy dispersion at normal criticality (blue) and multi-criticality (red) respectively.There occurs different dispersion around multicriticality.(c) Ground-state energy density which do not recognizes the spherical cap of the elliptical region.The peaks represent the criticalities ǫ = ±2t cos(φ) .(d-f) Dynamical, localization and crossover critical exponent at right side of a multi-critical point.The inset figures represent corresponding critical exponent towards the left side of the multi-critical point.

Figure 6 .
Figure 6.(a) Phase diagram of extended-range Hermitian model with broken TRS with φ = 3π/10 .There are two coupling neighbors, with parameter space t = α = 1 .The colored gapless region is ill-defined and topological quantities can not be defined in this region.(b) Energy dispersion at normal criticality (blue) and multi-criticality (red) respectively.There occurs different dispersion around Multi-criticality. (c) Groundstate energy density which do not recognizes the spherical cap of the elliptical region.The peaks represent the criticalities ǫ = ±2t cos(φ) .(d-f) Dynamical, localization and crossover critical exponent at left side of a multi-critical point.The inset figures represent corresponding critical exponent towards the right side of the multi-critical point.
Reports | (2024) 14:4504 | https://doi.org/10.1038/s41598-024-54946-5www.nature.com/scientificreports/The basis of the Bloch Hamiltonian can be rotated through the unitary generator U(n) .The Hamiltonian in new basis can be written as where χ ′ x , χ ′ y and χ ′ z are the modified winding vectors.The energy dispersion is given by

)Figure 8 .Figure 9 .
Figure 8. (a1-d1) Dynamic, crossover, localization and susceptibility critical exponents of short-range Hamiltonian at P T symmetry breaking point.(a2-d2) The insets represent the same critical exponents with absolute spectra in the P T broken regime.

Figure 10 .
Figure 10.Dynamic, crossover, localization and susceptibility critical exponents of long-range Hamiltonian in the regime α > 2 (a1-d1) at P T symmetry breaking point.(a2-d2) The insets represent the same critical exponents with absolute spectra in the P T broken regime.